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I am wondering about the definition of Power Spectral Density, in the context of measuring noise, i.e., Johnson-Nyquist (thermal) noise.

Usually a windowing method, like Welch's method, is used to derive the PSD. In this case, the PSD is the power spectrum divided by the frequency resolution times the window parameter, e.g., 1.5 for "Hanning."

But in the definitions of thermal noise, the bandwidth of the measurement is important. In order to arrive at a "thermal noise spectral density" one takes the noise power spectrum and divides by the measurement bandwidth, which is usually an analog bandpass filter built into the measurement apparatus.

So, should noise spectral density not be windowed? I am just really confused about how to arrive at the right value of the noise spectral density: divide by measurement bandwidth or use Welch's windowing method to get the "bandwidth?"

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I think your confusion comes from mixing continuous and discrete time cases. In the continous ("analog") case, you indeed have to divide by the bandwidth of the bandpass filter, else you would not get a spectral density but just power/energy. In the discrete time case, you divide by the frequency resolution, which is essentially the same thing, only instead of a bandwidth, you divide by the number of frequency bins.

Windowing is a different beast, it is applied to reduce spectral leakage, but changes the overall energy, which has to be corrected for by multiplying with the window specific correction factor. In general, this is only applied in the discrete case.

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  • $\begingroup$ I see. So in any real-world application (digitization of an analog signal), it will be the later case, i.e., dividing by the frequency resolution and not the bandwidth of the bandpass filter (which acts on the pre-digitized signal)? $\endgroup$
    – Nick
    Oct 10, 2023 at 11:32
  • $\begingroup$ Yes. In any case, you somehow have to put the frequency dimension in the denominator, otherwise you will not get a spectral density. $\endgroup$
    – Max
    Oct 10, 2023 at 12:06

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